Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,28,30,24)(20,25,34,42)(21,35,23,27)(29,45,46,38)(31,39,36,40)(32,48,47,41)(33,44,37,43);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,26,28)(20,32,34,47)(21,31,23,36)(25,41,42,48)(27,39,35,40)(29,37,46,33)(38,43,45,44);(10,14)(13,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,38)(26,42)(27,43)(28,45)(30,46)(31,47)(32,36)(35,44)(39,41)(40,48);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(18,26)(19,30)(25,42)(27,35)(29,46)(31,36)(32,47)(43,44);(17,21,22,23)(18,27,26,35)(19,31,30,36)(20,33,34,37)(24,39,28,40)(25,43,42,44)(29,47,46,32)(38,41,45,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418340);